Widening Perspectives

In this chapter we put forward points of interest from the perspective of future research, in seven sections: (1) The history of algebraic ideas. (2) The dialectics of syntax and semantics in the study of the generation of errors and in new points of interest concerning the representation of the unknown, based on something that is also unknown. This leads us to analyze the usual treatments for teaching systems of equations and to propose new ways of teaching them based on the use of trial and error, canceling, comparison, and substitution. (3) The study of the new problems posed by information and communication technology (ICT) by analyzing the new roles in the classroom or in the communication between learner and teacher. (4) An analysis of the MSS of Jordanus de Nemore’s De Numeris Datis, which makes it possible (a) to propose teaching models for second-degree equations; (b) to study the reason for the greater difficulty in the use of factorization compared with the learning of algebraic identities; and (c) to study the difficulties in using an algorithm as a subroutine in another more complex algorithm. (5) Early algebra, a rich new field for research with a huge past of research results. (6) Investigation of theoretical questions that involve the use of research into the instruction imparted by teachers. And (7), in the interests of (6), the need to develop the theoretical aspects of the communication model for the classroom using ICT.